Order Invariants Research Articles

Early analogue modeling experiments and related studies to today’s problems of geo-electromagnetic exploration

As I learned it from extensive geo-electromagnetic analogue modeling experiments, some specific nonconventional interpretation parameters, in certain conditions, give more detailed information about the geometry of subsurface resistivity inhomogeneities than the routinely used parameters. In this article, I show several examples, and I present how these early results influenced our later research. An enhanced geometric sensitivity may be due to special array geometry (as we call it “null array”), or it may be due to a narrow and very special frequency range (i.e., the so-called “keyhole” range). Nonconventional but physically based interpretation parameters (like the Poynting vector) or higher order invariants of resistivity or impedance tensors may also give useful additional information about the shape of subsurface bodies. One should be very careful in their application because a large part of these nontraditional approaches are strongly constrained by measuring errors and geological noise.

Explicit formulas for Vasil'ev invariants of the fourth order for knots

In this work, the integer-valued basic Vasil'ev invariants of the fourth order for the oriented knots V 4 1 (K) and V 4 2 (K) are presented. The explicit diagram-arrow formulas with rational coefficients defined according to the Gaussian diagram of a knot are obtained for them.

Invariants of finite order for braidings with symmetries

The author constructs a universal invariant for symmetric braids, knots, and linkages. CONTENTS

Differential invariants and group classification of variable coefficient generalized Gardner equation

By using Lie’s invariance infinitesimal criterion, we obtain the continuous equivalence transformations of a class of nonlinear Gardner equations with variable coefficients. Starting from the equivalence algebra， we construct the differential invariants of order one and make group classification. Finally some general class of variable coefficient nonlinear Gardner equations can be mapped to constant-coefficient mKdV equation and KdV-mKdV equation. In particular, some exact solutions of the Gardner equation with variable coefficients are obtained.

Immersions of surfaces into aspherical 3-manifolds

. We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions and classify all order 1 invariants.

Comment on ‘invariants of differential equations defined by vector fields’

In Ndogmo (2008 J. Phys. A: Math. Theor. 41 025207), the author claims to have determined a complete list of functionally independent differential invariants up to order 2 for the equivalence group of differential equations defined by vector fields when there are two and three independent variables. In this comment, we show that this is not the case. Using the equivariant moving frame method, we derive a complete set of functionally independent differential invariants of orders 1 and 2 for an arbitrary number n ⩾ 2 of independent variables. In the particular case n = 2, we obtain six functionally independent invariants, two of which were not found in Ndogmo (2008 J. Phys. A: Math. Theor. 41 025207). In the case n = 3, we get 21 functionally independent invariants, six of which are new. We also give a complete classification of the differential invariants.

Invariants of two- and three-dimensional hyperbolic equations

We consider linear hyperbolic equations of the form u t t = ∑ i = 1 n u x i x i + ∑ i = 1 n X i ( x 1 , … , x n , t ) u x i + T ( x 1 , … , x n , t ) u t + U ( x 1 , … , x n , t ) u . We derive equivalence transformations which are used to obtain differential invariants for the cases n = 2 and n = 3 . Motivated by these results, we present the general results for the n-dimensional case. It appears (at least for n = 2 ) that this class of hyperbolic equations admits differential invariants of order one, but not of order two. We employ the derived invariants to construct interesting mappings between equivalent equations.

Knot adjacency and fibering

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of knot adjacency can be used to obtain obstructions to the fibering of knots and of 3-manifolds. As an application, given a fibered knot $K’$, we construct infinitely many non-fibered knots that share the same Alexander module with $K’$. Our construction also provides, for every $n\in N$, examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order $\leq n$.

Contact Relative Differential Invariants for Non Generic Parabolic Monge-Ampère Equations

We find relative differential invariants of different orders for non generic parabolic Monge-Ampere equations (MAE?s). They are constructed in terms of some tensors associated with the derived flag of the characteristic distribution. The vanishing of such invariants allows one to determine the classes of each non generic parabolic MAE with respect to contact transformations.

Spectral invariants in Lagrangian Floer theory

Let $(M,\omega)$ be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold $L$ such that $\omega |_{\pi_2(M,L)}=0$ and $\mu|_{\pi_2(M,L)}=0$, where $\mu$ is the Maslov index. Given any Lagrangian submanifold $L'$, Hamiltonian isotopic to $L$, we define Lagrangian spectral invariants associated to the non zero homology classes of $L$, depending on $L$ and $L'$. We show that they naturally generalize the Hamiltonian spectral invariants introduced by Oh and Schwarz, and that they are the homological counterparts of higher order invariants, which we also introduce here, via spectral sequence machinery introduced by Barraud and Cornea. These higher order invariants are new even in the Hamiltonian case and carry strictly more information than the classical ones. We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.

On the Stability of a Class of Modified Gravitational Models

Motivated by the dark energy issue, a minisuperspace approach to the stability for modified gravitational models in a four dimensional cosmological setting is investigated. Specifically, after revisiting the f(R) case, R being the Ricci curvature, we present a stability condition around a de Sitter solution valid for modified gravitational models of generalized type F(R,G,Q), G and Q being the Gauss-Bonnet and quadratic Riemann invariants respectively. A generalization to higher order invariants is presented.

Irreducible forms for the metric variations of the action terms of sixth-order gravity and approximated stress–energy tensor

We provide irreducible expressions for the metric variations of the gravitational action terms constructed from the 17 curvature invariants of order six in the derivatives of the metric tensor, i.e., from the geometrical terms appearing in the diagonal heat-kernel or Gilkey–DeWitt coefficient a3. We then express, for a four-dimensional spacetime, the approximated stress–energy tensor constructed from the renormalized DeWitt–Schwinger effective action associated with a massive scalar field. We also construct, for higher dimensional spacetimes, the infinite counterterms of order six in the derivatives of the metric tensor appearing on the left-hand side of Einstein equations as well as the contribution associated with the cubic Lovelock gravitational action. In the appendix, we provide a list of geometrical relations we have used and which are more generally helpful for calculations in two-loop quantum gravity in a four-dimensional background or for calculations in one-loop quantum gravity in higher dimensional background. We also obtain the approximated stress–energy tensors associated with a massive spinor field and a massive vector field propagating in a four-dimensional background.

Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R 3 , with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T ⊕ P ⊕ Q , where T is a Z valued invariant reflecting the number of triple points of the immersion, and P , Q are Z / 2 valued invariants characterized by the property that for any regularly homotopic immersions i , j : F → R 3 , P ( i ) − P ( j ) ∈ Z / 2 (respectively, Q ( i ) − Q ( j ) ∈ Z / 2 ) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j. For immersion i : F → R 3 and diffeomorphism h : F → F such that i and i ○ h are regularly homotopic we show: P ( i ○ h ) − P ( i ) = Q ( i ○ h ) − Q ( i ) = ( rank ( h ∗ − Id ) + ε ( det h ∗ ∗ ) ) mod 2 where h ∗ is the map induced by h on H 1 ( F ; Z / 2 ) , h ∗ ∗ is the map induced by h on H 1 ( F ; Q ) , and for 0 ≠ r ∈ Q , ε ( r ) ∈ Z / 2 is 0 or 1 according to whether r is positive or negative, respectively.

Cn-MOVE AND ITS DUPLICATED MOVE OF LINKS

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, Cn-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by Cn-moves. We define a local move, βn-move, which is obtained from a Cn-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a βn-move and that a βn-move is realized by twice Cn-moves. In this note we study the relation between Cn-move and βn-move, and in particular, give answers to the following questions: (1) Is a βn-move realized by a finite sequence of Cn+1-moves? (2) Is Cn-move realized by a finite sequence of βn-moves?

A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations

Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, A method to tackle first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 3899–3910, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Analyzing the structure of the integrating factors for first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 1001–1006]. Apart from practical computational considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in the positive case, finds it via quadratures.

Higher dimensional black holes with a generalized gravitational action

We consider the most general higher-order corrections to the pure gravity action in $D$ dimensions constructed from the basis of the curvature monomial invariants of order 4 and 6, and degree 2 and 3, respectively. Perturbatively solving the resulting sixth-order equations we analyze the influence of the corrections upon a static and spherically symmetric back hole. Treating the total mass of the system as the boundary condition we calculate location of the event horizon, modifications to its temperature and the entropy. The entropy is calculated by integrating the local geometric term constructed from the derivative of the Lagrangian with respect to the Riemann tensor over a spacelike section of the event horizon. It is demonstrated that identical result can be obtained by integration of the first law of the black hole thermodynamics with a suitable choice of the integration constant. We show that reducing coefficients to the Lovelock combination, the approximate expression describing entropy becomes exact. Finally, we briefly discuss the problem of field redefinition and analyze consequences of a different choice of the boundary conditions in which the integration constant is related to the exact location of the event horizon and thus to the horizon defined mass.

Equivalence transformations and differential invariants of a generalized nonlinear Schrödinger equation

By using the Lie's invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear Schr\"{o}dinger equations with variable coefficients. Starting from the equivalence generators we construct the differential invariants of order one. We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schr\"{o}dinger equations which can be mapped, by means of an equivalence transformation, to the well known cubic Schr\"{o}dinger equation. We also provide the explicit form of the transformation.

The differential equation satisfied by a plane curve of degree n

Eliminating the arbitrary coefficients in the equation of a generic plane curve of order n by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much simpler when interpreted in terms of symmetric functions.

The QCD partition function in the external field in the covariant gauge

The QCD partition function in the external stationary gluomagnetic field is computed in the third order in external field invariants in arbitrary dimension and arbitrary covariant gauge. The contributions proportional to third order invariants in gluon field strength are shown to be dependent on covariant quantum gauge fixing parameter \alpha

First Order Semi‐Local Invariants of Stable Maps of 3‐Manifolds into the Plane

In the late 1980s, Vassiliev introduced new graded numerical invariants of knots, which are now called Vassiliev invariants or finite‐type invariants. Since he made this definition, many people have been trying to construct Vassiliev type invariants for various mapping spaces. In the early 1990s, Arnold and Goryunov introduced the notion of first order (local) invariants of stable maps. In this paper, we define and study first order semi‐local invariants of stable maps and those of stable fold maps of a closed orientable 3‐dimensional manifold into the plane. We show that there are essentially eight first order semi‐local invariants. For a stable map, one of them is a constant invariant, six of them count the number of singular fibers of a given type which appear discretely (there are exactly six types of such singular fibers), and the last one is the Euler characteristic of the Stein factorization of this stable map. Besides these invariants, for stable fold maps, the Bennequin invariant of the singular value set corresponding to definite fold points is also a first order semi‐local invariant. Our study of unstable fold maps with codimension 1 provides invariants for the connected components of the set of all fold maps. 2000 Mathematics Subject Classification 57R45 (primary), 32S20, 58K15 (secondary).